Representations of real reductive Lie groups

实数还原李群的表示

• 批准号: 10640153
• 负责人: MATUMOTO Hisayosi
• 金额: $ 1.02万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640153/
• 关键词: real reductive Lie groups; Unitary representations; degenerate series; derived functor modules; 退化系列表現; 既約ユニタリ表現; 分岐則; 半単純リー群; ホイタッカーモデル

In this academic year, I have been studied mainly on degenerate principal series of real reductive Lie groups and obtained the following. We consider a maximal parabolic subgroup of SO(m, n) (resp. U(m, n)) such that its Levi part is isomorphic to SO(m - n) x GL(n,R) (resp. U(m - n) x GL(n, C)). We consider the representations of SO(m, n) (resp, U(m, n)) induced from the representations of the parabolic subgroup coming from irreducible finite-dimensional representations of SO(m - n) (resp. U(m - n))) and one-dimensional representation of GL(n, R) (resp. GL(n, R)). In the last academic year, I found a reducibility of the representation obtained by considering the restriction to SO(m, l) (resp. U(m, 1)). In this year, I obtained an irreducibilty result. For the case of U(m, n) and the "sufficitintly" positive case" of SO(m, n), there is no reducibility other than the above. For the case of SO(m, n), the situation is quite subtle. In fact, Farmar had found an extra reducibility at the most singular parameter for the case of SO(3, 2).Our reducibility is described in terms of K-type decomposition of the degenerate principal series. It is compatible with the restriction tosmaller SO(m, k) (k < n) and we can obtain branching rule of some derived functor modules which appear as irreducible constituents.

在这个学年中，我主要研究了堕落的主要还原性谎言组，并获得了以下内容。我们考虑了SO（m，n）（分别u（m，n））的最大抛物线亚组，使其LEVI部分与SO（M -N）X GL（N，R）（u（m -n）（m -n）x GL（N）X GL（N）X GL（N，C））。我们考虑由SO（m -n）（m -n）（m -n）（resp。U（m -n））和GL（n，r）的一维表示（ressementional）表示的（m -n）（m -n）（m -n））的SO（m，n）（m，n）（resp，u（m，n））的SO（resp，u（m，n））的表示（so（m -n））（r）（n，r）（n，r）（resp.n，r）。在上一学年，我发现通过考虑限制SO（m，l）（分别u（m，1））获得的表示形式可降低。在今年，我获得了不核结果。对于u（m，n）和SO（m，n）的“正面的“积极情况”，除上述情况外都没有其他可理性。对于SO（m，m，m）而言，情况非常微妙。实际上，Farmar在SO中以最单一的参数（3，2）中最单一的参数中发现了一个额外的降低性（3，2）。与限制性TOSMALLER SO（M，K）（K <n）兼容，我们可以获得某些派生的函数模块的分支规则，这些模块显示为不可约组的成分。